Sergei Natanovich Bernstein (1880-1968)
-
Alternative spellings: Bernshtein, Bernshteyn
- Bernstein, S. N.,
Démonstration
du Théorème de Weierstrass fondée sur le calcul des
Probabilités,
Comm. Soc. Math. Kharkov 2.Series XIII No.1 (1912), 1-2.
This is Bernstein's famous paper where he presented a probabilistic proof
of the Weierstrass Theorem, and introduced what we today call Bernstein
polynomials. Note that his proof is somewhat "overinvolved". We nowadays
present this proof in a slightly more elegant form. This paper is
reprinted in Russian in Bernstein's collected works. Note that the bound volume
XIII of the journal carries the year 1913 even though the first few numbers
published separately each carry the year 1912.
- Bernstein, S. N., Sur l'ordre de la
meilleure approximation des fonctions continues par les polynômes de
degré donné, Mem. Cl. Sci. Acad. Roy. Belg.
4 (1912), 1-103. This paper was awarded a
prize by the Belgian
Academy of Science. This was as a consequence of his answer to a question
posed by de La Vallée Poussin. Bernstein proved that it is not
possible to approximate |x| in [-1,1] by a polynomial of
degree n with an approximation of order greater than 1/n. It
also contains the first form of what we call inverse theorems, Bernstein's
inequality and more.
- Bernstein, S. N.,
O nailuchshem priblizhenii nepreryvnykh funktsii posredstvom mnogochlenov dannoi
stepeni
Comm. Soc. Math. Kharkov 2.Series, XIII No. 2-5, (1912),
49-194, and
here is the somewhat changed version of the above paper, as it
appears in
Bernstein Collected Works, Constructive Function Theory 1905-1930,
Akademia Nauk SSSR, 1952, 11-104.
Note that the bound volume
XIII of the journal carries the year 1913 even though the first few numbers
published separately each carry the year 1912.
- Akhiezer, N. I., Academician S. N. Bernstein and his Work on
the Constructive Theory of Functions (in Russian), Izdat. Harkov.
Gosudarstv. Univ., Kharkov, 1955; translated into German as Das
Akademiemitglied S. N. Bernstein und seine Arbeiten zur konstruktiven
Funktionentheorie, translated and with a preface by Ralitza K.
Kovacheva and Heinz H. Gonska. in Mitt. Math. Sem. Giessen No. 240 (2000),
xiv + 97 pages. The original is an expanded form of the survey paper by N.
I. Akhiezer with the same title which appeared in Uspehi Mat. Nauk (N.S.)
6 (1951), no. 1(41), 3-67, which was written by Akhiezer on the occasion
of Bernstein's 70th birthday.
- Aleksandrov, P. S., N. I. Akhiezer, B. V. Gnedenko, and A.
N. Kolmogorov, Sergei Natanovich Bernstein: Obituary, (in Russian)
Uspekhi Mat. Nauk 24 (3) (1969), 211-218, translation in
Russia Math. Surveys 24 (3) (1969), 169-176.
- Garloff, J., A. P. Smith,
Preface, in Special Issue on
the Use of
Bernstein Polynomials in Reliable Computing: A Centennial Anniversary of
Reliable Computing 17 (2012), i-vi.
- Stark, E. L., Bernstein-Polynome 1912-1955, Functional
Analysis and Approximation (Oberwolfach 1980), Internat. Ser. Numer.
Math. 60, Birkhäuser, Basel-Boston, 1981, pp.443-461. Discusses
Bernstein's introduction of Bernstein polynomials in his 2-page paper on
Weierstrass' Theorem and the resulting literature, including the problem
of dating Bernstein's paper.
- Videnskii, V. S., Sergei Natanovich Bernshtein - Founder of the
constructive theory of functions. Uspehi Mat. Nauk 16 1961 no.
2 (98), 21?24. In English in Russian Math. Surveys 16 1961,
17-20.
See
here.
